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Math55_fa05quizzes_bergman

# Math55_fa05quizzes_bergman - Quiz 1 Problem 1(10 Describe a...

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Unformatted text preview: Quiz 1 Problem 1 (10) Describe a procedure for determining if the intersection of two ﬁnite sets ,say {a1,a2,...,an} and {171,152, Wing}, is a subset of another ﬁnite sets, say {c1,...,cm}. Express the procedure in a pseudocode. Count the number of steps it takes for the program to halt. W94“ sew? (hi-“M 7 i é»~~,i3, mes) A' i 422 W3 f2: 1:) "5) 27, 7’, "z “3 M) Z //1 C -’ 7 f,'g.>/)UJ)1 ,7 JOMW ﬁéggé? (€‘J/"')d"z2{er‘Jéhg/z Cu‘ ’2']: {A’J'UL'KS (1:1 {(1) “V”; [M3 a) [,3 cvn‘ébii’h'} 4355th ﬁts ‘ [FM (I?! +9 *1 V f? a-ee A 4; ¢ 5 "LL caste/5”"? ’ | m V” J Ant-r :6 5-1“ AM 5? c, gr; an Vol...) Quiz 3 Problem 1 (5) Prove that if f is 0(9) and g is 0(h) then f is 0(h). (Write this in English sentences. Explain the meaning of each symbol that you will be using, and for each symbol explain - how or why one can get the object denoted by the symbol). (e063? Agfofé? “*7?€0CA§ ‘ K'er 296T?" Fa-nﬁaxq ['j J: a : ‘Ef __ X>/L, old %jchA x>lzl_wlf1’l W’ ﬂ: is Assam” we l 4M 4 MM 0, “A” we 7,4» 21“ MM» AP CL. / 1c. a 5. z (,2, ﬂ sag CfA ha 1% mow ﬁlnwm. 70 ‘ C fw‘w} «[3 i 0C1) lulu”. L174 +fuk LAD/(1)441} Wlékpﬁﬁ n‘ W 1/4319 ‘ . .G i O (5/ M jAK 27177:, JAM 4 “Fox” VelmélxvlL’t’arLjrﬂJ/M {UM doxﬁ'hir. SEWLé/ﬂp “w jaw/5 All 5:065); Twy/g/ -P )3 "4/5 )Lu m 47,“! 4'3 53k, ﬂu gﬁw’wA ml‘zm [3' \$55 “372% Alm; {M tea/ﬁ— .< (7% 90% 4' 445\$) 74%; A/ WEI: LS‘; l‘Q—nv, Q’r J/i Problem 2 (5) Prove that iff : S —+ Y and if X, Z are subsets of S, then f(X U Z) : f(X) U f {Write this in Engh'sh sentences. Explain the meaning of each symbol that you will be using, and for each symbol explain how or why one can get the object denoted by the symbol). a? . SEW, ﬂick; e (We a a me "x 4 W") Mg Y"? array fl“ or FM {LA {S ’7‘ 4M {Sara-Al}, 5? Alia] 1'? r is M WI’H‘x £511) is m ffy)‘ Inc X131“ aﬁga) is _/A -F(—Z—). . t o/ ﬂnggc’lgL 5%,ch a? an; "L inc “Fa/{7‘ (Liam, 47? 5/ X) 4V0 is A chub—Fm) M m ¥(?U75). "L? X {’5 A web 3( or 27, 100K 4” X/ 940 z; 2% 01514:! ’{ “F179, arm MM é,” “PL Mei. (mum mm A 465”! 2;, regime can ﬂy wen/1% 4H M 1"? [S [*3 1C AU?) ﬁnd a" ﬁlm )CCAU}):.FCX)U—PIZU ‘ J?§\ ig‘mw No [736% Quiz 4 COMMENT: This quiz is out 0f 20 points. Your grade for this quiz will be divided by 2, to, make sure that all quizzes or out of 10. Problem 1 (5) Using Euclidian Algorithm ﬁnd the greatest common divisor of 385 and 924. 3‘55 cave-w u :3v5-2 % , ‘7’” ’5“ ya! (moo =77 . 15H 4?; 3?: ‘7 z + 7‘7 23%”; 1222 Problem 2 (15) Shoe that if}? is a. prime number then either 4mg 5 1(modp) has no solutions or the only solutions of it are integers 1: such that a: E 1J—jg—1(n10d;o) or x E P—Elﬁnodp). [Hint This is like the homework problem. First try to ﬁgure out for which p’s there are no solutions. Then show that for other p’s all solutions have to be of the given form. The last step is to show that every x of the given form is a solution. You can use the back of this paper] . ,, 40:12; \ c J"; )0 2’3 Prim “T? (fat I 5-: A; k 5a)“; V é: L Q'Jto/ l/ ‘X " (J 4“ A0? Kali-514W "‘F p fax/Lg, (ff-Ix UL? {we +L Agar: in, 112” x a, 17.417196! N P [is 0" 0J0! “MAM‘ M (Rx/5+; he 51 5 ASSQM, ah ON # a“ I41. 904%". x; A; Par». Fe; = L2 is 6% Mfg; ,( J 1 gr? .m—n- 1 OJ )4 [1 at." 1 +17 2 f; Gtzj’gﬂ'ﬂ/u‘hnf. 7%. ‘ Flt,ng ’97"? “is/2"] ammﬁh/ Z \ / nﬁsém Quiz 5 Problem 1 (5) Using Fermat’s Little T hearem, Show that 368 E 13(m0d17). Pr} 474,39 t7 q 3/ fall/a 3r, :1 (Y to 1.25"“! % 99:;(MP) S n {bl/36 ,ﬂ q, 25,;qu p-1 1:733 3 ‘3 EI’SKAHIV) 5(«(7% ’€ 7 l3 “3“”? I? Problem 2 { 5 ) Prove that there are inﬁnitely many primes of the form 616 + 5, and explain why ' doesn’t the same proof imply that there are inﬁnitely many primes of the form 616 + 1. atpﬂagy {[7 7 f), {an [A d t &I/ [army/5 a) 4 {ﬁlch'jnt ‘va [{1“; :[(&+I) —/ MM ‘ 1/ WM 29 47a». 512.73“ ff K545”. the; in 6: (Jélwﬂ {4m} .wKMd/V: 5 K Wait/iv" «ﬂatten <52 amtew/ 54 74/ Vi Hwy ...
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